In my opinion, there is a difference between the concepts of “Art” and “Design.”
To begin, I don’t believe that “Design” and “Art” are linked in their concept. To me, “Design” is purpose-driven in its origin. While “Art” may be created purposefully, Art’s intent is often perceived as a secondary asset. I would wager that most people would think that Art’s meaning is subjective and mutable. By nature, Design is intended to control or adapt to specific needs or ideas.
In that sense, Art and Design are different because Art is less constrained by definition. When working artistically, meaning and intent may not be factors in the creative process. When designing, intent and purpose are critically important. Art can be tailored by design, and design can be executed artfully, but the two are related only tangentially in that sense. “Every square is a rectangle, not every rectangle is a square.”
Visual mediums are linked in this respect. You can often use the same descriptors to describe them, but the fact that they share descriptors is not proof that they are concretely linked.
Design is a process by which we may generate Art. In the most literal sense, you may perceive any and all things as “Designed.” With talent and sufficient time, anyone could replicate the Mona Lisa. However, behind all great Art is the creative intent and an unrepeatable vision. It is likely that you could design something in the same way that it was conceived originally. However, Art begets design and re-design while at the same time being created by the same processes by which it is replicated.
In the end, Design and Art are related in many ways but their differences (however minute you may perceive them) separate them greatly. Behind Art and Design are intent, but Art is distanced from Design by its perceived “originality.”